Abstract
We study the entanglement entropy of gauged internal degrees of freedom in a two dimensional symmetric product orbifold CFT, whose configurations consist of N strands sewn together into “long” strings, with wavefunctions symmetrized under permutations. In earlier work a related notion of “entwinement” was introduced. Here we treat this system analogously to a system of N identical particles. From an algebraic point of view, we point out that the reduced density matrix on k out of N particles is not associated with a subalgebra of operators, but rather with a linear subspace, which we explain is sufficient. In the orbifold CFT, we compute the entropy of a single strand in states holographically dual in the D1/D5 system to a conical defect geometry or a massless BTZ black hole and find a result identical to entwinement. We also calculate the entropy of two strands in the state that represents the conical defect; the result differs from entwinement. In this case, matching entwinement would require finding a gauge-invariant way to impose continuity across strands.
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References
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
G. Vidal, J.I. Latorre, E. Rico and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90 (2003) 227902 [quant-ph/0211074] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
D. Gioev and I. Klich, Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture, Phys. Rev. Lett. 96 (2006) 100503 [INSPIRE].
J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333 [INSPIRE].
S.W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
V. Balasubramanian, M.B. McDermott and M. Van Raamsdonk, Momentum-space entanglement and renormalization in quantum field theory, Phys. Rev. D 86 (2012) 045014 [arXiv:1108.3568] [INSPIRE].
C. Agon, V. Balasubramanian, S. Kasko and A. Lawrence, Coarse Grained Quantum Dynamics, Phys. Rev. D 98 (2018) 025019 [arXiv:1412.3148] [INSPIRE].
L. Susskind and E. Witten, The Holographic bound in anti-de Sitter space, hep-th/9805114 [INSPIRE].
L. Susskind, Holography in the flat space limit, AIP Conf. Proc. 493 (1999) 98 [hep-th/9901079] [INSPIRE].
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: A Conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
V. Balasubramanian, R. Gopakumar and F. Larsen, Gauge theory, geometry and the large N limit, Nucl. Phys. B 526 (1998) 415 [hep-th/9712077] [INSPIRE].
J. Polchinski, M theory and the light cone, Prog. Theor. Phys. Suppl. 134 (1999) 158 [hep-th/9903165] [INSPIRE].
V. Balasubramanian, A. Bernamonti, B. Craps, T. De Jonckheere and F. Galli, Entwinement in discretely gauged theories, JHEP 12 (2016) 094 [arXiv:1609.03991] [INSPIRE].
V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [INSPIRE].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Matrix string theory, Nucl. Phys. B 500 (1997) 43 [hep-th/9703030] [INSPIRE].
S. Ghosh, R.M. Soni and S.P. Trivedi, On The Entanglement Entropy For Gauge Theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].
R.M. Soni and S.P. Trivedi, Aspects of Entanglement Entropy for Gauge Theories, JHEP 01 (2016) 136 [arXiv:1510.07455] [INSPIRE].
H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
W. Donnelly, Decomposition of entanglement entropy in lattice gauge theory, Phys. Rev. D 85 (2012) 085004 [arXiv:1109.0036] [INSPIRE].
W. Donnelly, Entanglement entropy and nonabelian gauge symmetry, Class. Quant. Grav. 31 (2014) 214003 [arXiv:1406.7304] [INSPIRE].
D. Radicevic, Notes on Entanglement in Abelian Gauge Theories, arXiv:1404.1391 [INSPIRE].
K. Van Acoleyen, N. Bultinck, J. Haegeman, M. Marien, V.B. Scholz and F. Verstraete, The entanglement of distillation for gauge theories, Phys. Rev. Lett. 117 (2016) 131602 [arXiv:1511.04369] [INSPIRE].
J. Schliemann, D. Loss and A.H. MacDonald, Double-occupancy errors, adiabaticity, and entanglement of spin qubits in quantum dots, Phys. Rev. B 63 (2001) 085311 [cond-mat/0009083].
J. Schliemann, J.I. Cirac, M. Kus, M. Lewenstein and D. Loss, Quantum correlations in two-fermion systems, Phys. Rev. A 64 (2001) 022303.
R. Paškauskas and L. You, Quantum correlations in two-boson wave functions, Phys. Rev. A 64 (2001) 042310 [quant-ph/0106117].
K. Eckert, J. Schliemann, D. Bruß and M. Lewenstein, Quantum correlations in systems of indistinguishable particles, Annals Phys. 299 (2002) 88 [quant-ph/0203060].
J. Lin, A Toy Model of Entwinement, arXiv:1608.02040 [INSPIRE].
S. Ghosh and S. Raju, Quantum information measures for restricted sets of observables, Phys. Rev. D 98 (2018) 046005 [arXiv:1712.09365] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP 07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
P. Lévay, S. Nagy and J. Pipek, Elementary formula for entanglement entropies of fermionic systems, Phys. Rev. A 72 (2005) 022302 [quant-ph/0501145].
P. Lévay and P. Vrana, Three fermions with six single-particle states can be entangled in two inequivalent ways, Phys. Rev. A 78 (2008) 022329 [arXiv:0806.4076].
L. Chen, D.Z. Djokovic, M. Grassl and B. Zeng, Four-qubit pure states as fermionic states, Phys. Rev. A 88 (2013) 052309 [arXiv:1309.0791] [INSPIRE].
G. Sárosi and P. Lévay, Entanglement classification of three fermions with up to nine single-particle states, Phys. Rev. A 89 (2014) 042310 [arXiv:1312.2786] [INSPIRE].
G. Sárosi and P. Lévay, Coffman-kundu-wootters inequality for fermions, Phys. Rev. A 90 (2014) 052303 [arXiv:1408.6735].
D. Harlow, The Ryu-Takayanagi Formula from Quantum Error Correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
V. Balasubramanian, J. de Boer, E. Keski-Vakkuri and S.F. Ross, Supersymmetric conical defects: Towards a string theoretic description of black hole formation, Phys. Rev. D 64 (2001) 064011 [hep-th/0011217] [INSPIRE].
V. Balasubramanian, P. Kraus and M. Shigemori, Massless black holes and black rings as effective geometries of the D1-D5 system, Class. Quant. Grav. 22 (2005) 4803 [hep-th/0508110] [INSPIRE].
M. Cvetič and F. Larsen, Near horizon geometry of rotating black holes in five-dimensions, Nucl. Phys. B 531 (1998) 239 [hep-th/9805097] [INSPIRE].
L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261 (1985) 678 [INSPIRE].
O. Lunin and S.D. Mathur, Correlation functions for M N /S N orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
F. Larsen and E.J. Martinec, U(1) charges and moduli in the D1-D5 system, JHEP 06 (1999) 019 [hep-th/9905064] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
G. Wong, I. Klich, L.A. Pando Zayas and D. Vaman, Entanglement Temperature and Entanglement Entropy of Excited States, JHEP 12 (2013) 020 [arXiv:1305.3291] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York (1997) [DOI:https://doi.org/10.1007/978-1-4612-2256-9].
J. Cardy, Thermalization and Revivals after a Quantum Quench in Conformal Field Theory, Phys. Rev. Lett. 112 (2014) 220401 [arXiv:1403.3040] [INSPIRE].
G. Mandal, R. Sinha and N. Sorokhaibam, Thermalization with chemical potentials and higher spin black holes, JHEP 08 (2015) 013 [arXiv:1501.04580] [INSPIRE].
P. Basu, D. Das, S. Datta and S. Pal, Thermality of eigenstates in conformal field theories, Phys. Rev. E 96 (2017) 022149 [arXiv:1705.03001] [INSPIRE].
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Balasubramanian, V., Craps, B., De Jonckheere, T. et al. Entanglement versus entwinement in symmetric product orbifolds. J. High Energ. Phys. 2019, 190 (2019). https://doi.org/10.1007/JHEP01(2019)190
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DOI: https://doi.org/10.1007/JHEP01(2019)190